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Câu hỏi: Phương trình ${{\log }_{49}}{{x}^{2}}+\dfrac{1}{2}{{\log }_{7}}{{\left( x-1 \right)}^{2}}={{\log }_{7}}\left( {{\log }_{\sqrt{3}}}3 \right)$ có bao nhiêu nghiệm?
A. 2.
B. 3.
C. 1.
D. 4.
A. 2.
B. 3.
C. 1.
D. 4.
Điều kiện $\left\{ \begin{aligned}
& x\ne 0 \\
& x\ne 1 \\
\end{aligned} \right..$
${{\log }_{49}}{{x}^{2}}+\dfrac{1}{2}{{\log }_{7}}{{\left( x-1 \right)}^{2}}={{\log }_{7}}\left( {{\log }_{\sqrt{3}}}3 \right)\Leftrightarrow {{\log }_{7}}\left| x \right|+{{\log }_{7}}\left| x-1 \right|={{\log }_{7}}2$
$\Leftrightarrow {{\log }_{7}}\left| x\left( x-1 \right) \right|={{\log }_{7}}2\Leftrightarrow \left[ \begin{aligned}
& x\left( x-1 \right)=2 \\
& x\left( x-1 \right)=-2 \\
\end{aligned} \right.\Leftrightarrow \left[ \begin{aligned}
& {{x}^{2}}-x-2=0 \\
& {{x}^{2}}-x+2=0 \\
\end{aligned} \right.\Leftrightarrow \left[ \begin{aligned}
& x=2 \\
& x=-1 \\
\end{aligned} \right..$
& x\ne 0 \\
& x\ne 1 \\
\end{aligned} \right..$
${{\log }_{49}}{{x}^{2}}+\dfrac{1}{2}{{\log }_{7}}{{\left( x-1 \right)}^{2}}={{\log }_{7}}\left( {{\log }_{\sqrt{3}}}3 \right)\Leftrightarrow {{\log }_{7}}\left| x \right|+{{\log }_{7}}\left| x-1 \right|={{\log }_{7}}2$
$\Leftrightarrow {{\log }_{7}}\left| x\left( x-1 \right) \right|={{\log }_{7}}2\Leftrightarrow \left[ \begin{aligned}
& x\left( x-1 \right)=2 \\
& x\left( x-1 \right)=-2 \\
\end{aligned} \right.\Leftrightarrow \left[ \begin{aligned}
& {{x}^{2}}-x-2=0 \\
& {{x}^{2}}-x+2=0 \\
\end{aligned} \right.\Leftrightarrow \left[ \begin{aligned}
& x=2 \\
& x=-1 \\
\end{aligned} \right..$
Đáp án A.